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\begin{equation}  \mathcal{M} = \frac{H}{r} = \frac{v_K}{c^{\rm{iso}}_s}  \end{equation}  Now we can rewrite (\ref{dP_ad}) and (\ref{dP_iso}) in terms of the disk Mach number  \begin{equation}  \int{\frac{dP}{\rho}} = \frac{(c^{\rm{ad}}_{s})^2 }{\gamma -1} = \frac{\gamma (c^{\rm{iso}}_{s})^2 }{\gamma -1} \quad \rm{Adiabatic} \ \rm{Flow}  \end{equation}  \begin{equation}  \int{\frac{dP}{\rho}} = (c^{\rm{iso}}_{s})^2 \ln{\frac{\rho}{\rho_0}} \quad \rm{Isothermal} \ \rm{Flow}  \end{equation}